Progress In Electromagnetics Research, PIER 89, 339 357, 29 SSOR PRECONDITIONED INNER-OUTER FLEXIBLE GMRES METHOD FOR MLFMM ANALYSIS OF SCAT- TERING OF OPEN OBJECTS D. Z. Ding, R. S. Chen, and Z. H. Fan Department of Communication Engineering Nanjing University of Science and Technology Nanjing 2194, China Abstract To efficiently solve large dense complex linear system arising from electric field integral equations (EFIE) formulation of electromagnetic scattering problems, the multilevel fast multipole method (MLFMM) is used to accelerate the matrix-vector product operations. The inner-outer flexible generalized minimum residual method () is combined with the symmetric successive overrelaxation (SSOR) preconditioner based on the near-part matrix of the EFIE in the inner iteration of to speed up the convergence rate of iterative methods. Numerical experiments with a few electromagnetic scattering problems for open structures are given to demonstrate the efficiency of the proposed method. 1. INTRODUCTION Electromagnetic integral equations are often discretized with the method of moments (MoM) [1 7], one of the most widespread and generally accepted techniques for electromagnetic problems. The formulation considered in this paper is the electric field integral equation (EFIE) as it is the most general and does not require any assumption about the geometry of the object. It is convenient to model objects with arbitrary shape using triangular patches; hence, RWG functions [8] are widely used to represent unknown current distributions. When iterative solvers are used to solve the MoM matrix equation, the fast multipole method (FMM) or multilevel fast Corresponding author: D. Z. Ding (dzding@mail.njust.edu.cn). Also with State KeyLaboratoryof Millimeter Waves, Southeast University, Nanjing 2196, P. R. China.
34 Ding, Chen, and Fan multipole method (MLFMM) [9 15] can be used to accelerate the calculation of matrix vector multiplications. The system matrix resulted from EFIE is often an ill-conditioned matrix and results in the low convergence of the Krylov iterative method [16]. Iterative methods for solving linear systems are usually combined with a preconditioner that can be easily solved. For some practical problems, however, a natural and efficient choice of preconditioner may be one that cannot be easily solved by a direct method and thus may require an iterative method (called inner iteration) itself to solve the preconditioned equations. There also exist cases where the matrix operator contains inverses of some other matrices, an explicit form of which is not available. Then the matrixvector multiplication can be obtained approximately through an inner iteration. The domain decomposition preconditioners [17], and some positive definite non-symmetric linear systems preconditioned by their symmetric parts are such examples. For these types of problems, the original iterative method will be called the outer iteration and the iterative method used for solving the preconditioner or forming the matrix-vector multiplication will be called the inner iteration. In order to be able to enhance robustness of iterative solvers, we should be able to determine whether or not a given preconditioner is suitable for the problem at hand. If not, one can attempt another possible iterative method/preconditioner and switch periodically if necessary. It is desirable to be able to switch within the outer iteration instead of restarting. For the generalized minimum residual method (GMRES) algorithm, this can be easily accomplished with the help of a rather simple modification of the standard algorithm, referred to as the flexible GMRES () [18 21]. An important property of is that it satisfies the residual norm minimization property over the preconditioned Krylov subspace just as in the standard GMRES algorithm. The method with FFT technique is applied for the analysis of electromagnetic wave scattering from threedimensional dielectric bodies with the large permittivity [19]. Moreover, the inner-outer flexible GMRES method can be combined with a preconditioner to further improve convergence as the conventional GMRES [18]. Simple preconditioners like the diagonal or diagonal blocks of the coefficient matrix can be effective only when the matrix has some degree of diagonal dominance [22]. Incomplete LU (ILU) factorizations have been successfully used for nonsysmmetric dense systems [23]. But the factorization is often very ill-conditioned that makes the triangular solvers highly unstable and the use of the ILU preconditioner may be totally ineffective [24]. Sparse approximate inverse (SAI) preconditioning techniques have been successfully used
Progress In Electromagnetics Research, PIER 89, 29 341 with the MLFMM [25, 26], however, the construction cost is usually very high. The symmetric successive over-relaxation (SSOR) [18, 27] preconditioner has the advantage of very small construction cost, which makes it preferable over general-purpose preconditioners such as ILU and SAI preconditioner techniques. Furthermore, SSOR preconditioning technique contains more information of the coefficient matrix when compared with a diagonal/block diagonal matrix, which is perhaps efficient only for very long and narrow structures. In this investigation, the SSOR preconditioning scheme is used to accelerate the inner-outer method, which is proposed to speed up the convergence of GMRES algorithm [18]. The symmetric successive overrelaxation (SSOR) preconditioner is constructed based on the nearpart matrix of the EFIE impedance matrix in the inner iteration of. Numerical examples are given to demonstrate the accuracy and efficiency of the SSOR preconditioned inner-outer algorithm in radar cross section (RCS) calculations This paper is outlined as follows. Section 2 gives an introduction of EFIE formulation combined with MLFMM. Section 3 describes the details to construct the SSOR preconditioned inner-outer algorithm based on the near-field matrix in MLFMM implementation. Numerical experiments with a few electromagnetic wave scattering problems are presented to demonstrate the efficiency of the SSOR preconditioned algorithm in Section 4. Section 5 gives some conclusions and comments. 2. EFIE FORMULATION AND MLFMM The EFIE formulation of electromagnetic wave scattering problems using planar Rao-Wilton-Glisson (RWG) basis functions for surface modeling is presented in [1]. The resulting linear systems from EFIE formulation after Galerkin s testing are briefly outlined as follows: where and N A mn I n = b m, m =1, 2,...,N (1) n=1 Z mn = jk f m (r) (Ī + 1 ) s s k 2 [G(r, r )f n (r )]dsds (2) b m = 1 η s f m (r) E i (r)ds, G(r, r )= e jk r r 4π r r (3)
342 Ding, Chen, and Fan where G(r, r ) refers to the Green s function in free space and {I n } is the column vector containing the unknown coefficients of the surface current expansion with RWG basis functions f m. Also, as usual, r and r denote the observation and source point locations. E i (r) is the incident excitation plane wave, and η and k denote the free space impendence and wave number, respectively. Once the matrix equation (1) is solved by numerical matrix equation solvers, the expansion coefficients {I n } can be used to calculate the scattered field and RCS. In the following, we use A to denote the coefficient matrix in equation (1), x = {I n }, and b = {b m } for simplicity. Then, the EFIE matrix equation (1) can be symbolically rewritten as: Ax = b (4) The basic idea of the fast multipole method (FMM) is to convert the interaction of element-to-element to the interaction of group-togroup. Here a group includes the elements residing in a spatial box. The mathematical foundation of the FMM is the addition theorem for the scalar Green s function in free space. Using the FMM, the matrixvector product Ax can be written as: Ax = A N x + A F x (5) where A N is the near part of A and A F is the far part of A. In the FMM, the calculation of matrix elements in A N remains the same as in the method of moments (MoM) procedure. However, those elements in A F are not explicitly computed and stored. Hence they are not numerically available in the FMM. It has been shown that the operation complexity of FMM to perform Ax is O(N 1.5 ). If the FMM is implemented in multilevel, the total cost can be reduced further to O(NlogN) [9]. 3. SSOR PRECONDITIONED INNER-OUTER Consider the iterative solution of large linear systems of equations as shown in (4). Let x C n be an initial guess for this linear system, r = b Ax be its corresponding residual and M 1 the right preconditioner. The Krylov subspace algorithm with right preconditioning solves the modified system: AM 1 (Mx)=b (6) Clearly the matrix AM 1 need not be formed explicitly: we only need to solve Mz = v whenever such an operation is required. In some cases, solving a linear system with the matrix M consists of forming an
Progress In Electromagnetics Research, PIER 89, 29 343 approximate solution by performing one or a few steps of a relaxation type method, or a few Chebyshev iterations. It is natural that the preconditioners is used not only a single step during an outer iteration, but as many as are needed to solve a linear system within a given tolerance. The preconditioner may be no longer constant but is allowed to vary from one step to another in the outer iteration. A similar situation in which the preconditioner is not constant is when another Krylov subspace method is used as a preconditioner. These lead us to raise the question of whether or not it is possible to accommodate such variations in the preconditioners and still obtain an algorithm that satisfies an optimality property similar to the one satisfied by the original iterative method. This question has been avoided in the past because in the Hermitian case there does not seem to have a version of the usual preconditioned conjugate gradient algorithm that satisfies a short vector recurrence and that allows the preconditioner to vary at each step. In the non-hermitian case and for methods that do not rely on short vector recurrences, such as GMRES, variations in the preconditioner can be handled without difficulty. The preconditioner M is constant in conventional preconditioned GMRES algorithm [18]. However, in Flexible GMRES () [18], the preconditioner is no longer constant but is allowed to vary from one step to another in the outer iteration. The algorithm is classically described by (1) Start: Choose x and a dimention m of the Krylov subspaces. Define an (m +1) m matrix H m and initialize all its entries h i,j to zero. (2) Arnoldi process: (a) compute r = b Ax, β = r 2 and v 1 = r /β. (b) for j =1,...,m do compute z j := Mj 1 v j ; compute w := Az j ; For i =1,...,j,do h i,j := (w, v j ) w := w h i,j v i Enddo compute h j+1,j = w 2 and v j+1 = w/h j+1,j Enddo (c) Define Z m := [z 1,...,z m ]. (3) Form the approximate solution: Compute x m = x + Z m y m βe1 where y m = arg min y H m y and e 1 =[1,,...,] T. 2
344 Ding, Chen, and Fan (4) Restart: If satisfied stop, else set x m x and go to 2. As can be observed, the only difference from the standard GMRES version is that we now save the preconditioned vector z j and update the solution using these vectors. Note that we can define z j in step (2) without reference to any preconditioner, i.e., we can simply pick a given new vector z j. In algorithm, the preconditioning operation z j := Mj 1 v j can be thought of as a means of approximately solving the matrix system M j z j = v j (7) where Mj 1 is the preconditioner. This is referred to as an inner iteration. In this investigation, the preconditioner M is taken to be the near part impedance matrix A N. For the inner iteration any method can be used, though for practical purposes it should be a fast approximate solve. In conventional algorithm, the GMRES algorithm is used for the preconditioning, or inner iterations. In this paper, we have SSOR preconditioned GMRES for the inner iterations, and as the (outer) flexible method, denoted as SSOR-. In the SSOR preconditioning scheme [27], the SSOR preconditioned inner iteration matrix system is as the following form M SSOR A N z j = v j (8) The SSOR preconditioner M SSOR is constructed based on the near part impedance matrix A N as the following form ( ) 1 ( M SSOR = D + L )( D D + U) (9) where A N = L + D + U in (9), L is the lower triangular matrix A N, D is the diagonal matrix of A N, U is the upper triangular matrix A N, and D =(1/ω)D, <ω<2(ω is the relaxation parameter). The additional cost needed by the SSOR- algorithm over the conventional algorithm is only in the extra memory required to save the SSOR preconditioner M SSOR in the inner iteration. On the other hand, the added advantage of faster convergence may certainly be worth this extra cost. There are a few applications in which this flexibility can be quite helpful, especially in the context of developing robust iterative methods or for developing preconditiomers for massively parallel computers. In the next section, the convergence of the SSOR- algorithm combined with MLFMM technique is discussed in detail for electromagnetic scattering by open conducting conductors.
Progress In Electromagnetics Research, PIER 89, 29 345 4. NUMERICAL EXPERIMENTS In this section, we show some numerical results for open conducting structures that illustrate the effectiveness of the proposed SSOR preconditioned inner-outer (SSOR-) for the solution of the EFIE linear systems in electromagnetic scattering problems. The EFIE linear systems based on the RWG basis functions are solved with MLFMM accelerated Krylov iterative methods. All numerical experiments are performed on a Pentium 4 with 2.9GHz CPU and 2 GB RAM in single precision. The restarted version of GMRES(m) algorithm is used as iterative method, where m is the dimension size of Krylov subspace for GMRES. Additional details and comments on the implementation are given as follows: Zero vector is taken as initial approximate solution for all examples and all systems in each example The iteration process is terminated when the normalized backward error is reduced by 1 3 for all the examples. m = 3 is used as the dimension of the Krylov subspace for the restarted GMRES algorithms. The inner stop precision is taken to be 1 1 in both the and SSOR- algorithms. We investigated the performance of the presented SSOR- algorithm on three open conducting structures and one closed structures. The first example is a metallic open-cone, which Monostatic RCS (db) 3 2 1-1 MLF MM BOR -2 3 6 9 12 15 18 θ (Degree) Figure 1. The monostatic RCS for vertical polarization at 3. GHz for an open cone.
346Ding, Chen, and Fan has a height of 2 cm and a base whose diameter is also 2 cm. The open cone is discretized with 1922 planar triangular patches for RWG basis functions with 2851 unknowns at 3 GHz. The second example is a closed missile structure, which is discretized with 5212 planar triangular patches for RWG basis functions with 7818 unknowns with 7818 unknowns at 2 MHz. The three example is a 2.5λ 2.5λ 3.75λ open-cavity with 1568 unknowns for RWG basis functions at 3 MHz (λ is wavelength in free space). The last example is a box-plate perfectly electrically conducting (PEC) scatterer consisting of a plate of size (1 m 1 m) placed on an 2 m 1 m large plate having a thickness of.3 m. The box-plate is discretized with 23232 planar triangular patches for RWG basis functions with 34784 unknowns at 5 GHz. The MLFMM was employed for the computation of the matrix/vector products involving non-near zone interaction elements in the computation. As shown in Figure 1, the monostatic vertical polarized RCS curve of the cone at 3. GHz is compared with that of a BOR (Body of Revolution) analysis. The size dimension of the group on the finest level is.2 wavelength (.2λ) and the MLFMM with two levels was employed for the cone. It can be found that the results using the MLFMM are in good agreement with that of the BOR analysis [28]. The monostatic vertical polarized RCS curves of the missile example at 2 MHz and the cavity example at 3 MHz are given in Figure 2 and Figure 3, respectively. The size dimension of the group on the finest level is quarter of the wavelength (.25λ) and the MLFMM with three levels was employed for both the missile and Monstatic RCS(dB) 25 2 15 1 5-5 MLFMA reference[29] -1 3 6 9 12 15 φ (degree) 18 Figure 2. The monostatic RCS for vertical polarization at 2 MHz for a missile.
Progress In Electromagnetics Research, PIER 89, 29 347 35 RCS(dB) 3 25 2 15 1 5-5 -1 MLFMM Reference [3] 2 4 6 8 1 12 14 16 18 θ (degree) Figure 3. The monostatic RCS for vertical polarization at 3 MHz for an open cavity. 7 6 d = d =.1m 5 RCS(dB) 4 3 2 1 2 4 6 8 1 12 14 16 18 θ (degree) Figure 4. The bistatic RCS at 5 GHz for a box-plate example. the cavity examples. Figure 4 gives the numerical results of Bistatic RCS for box-plate example. For this example, the size dimension of the group on the finest level is.5 wavelength (.5λ) and the MLFMM with 5 levels was employed. To verify the accuracy of the method, a closed scatterer consisting of a plate of size (1 m 1 m) having a thickness of.1 m placed on the above plate is considered in Figure 4. It can be obseverd that our results agree well with those in [29, 3]. As shown in Figures 5 7, the convergence histories of proposed SSOR- algorithm for system 1 are given for the first three
348 Ding, Chen, and Fan Residual Norm 1.E+ 1.E-1 1.E-2 GMRES SSOR-GMRES SSOR- 1.E-3 2 4 6 8 1 Number of Matrix-vector Multiplications Figure 5. Convergence history of GMRES algorithms for system 1 on the open-cone example. Residual Norm 1.E+ 1.E-1 1.E-2 GMRES SSOR-GMRES SSOR- 1.E-3 6 12 18 24 3 36 42 48 54 6 Number of Matrix-vector Multiplications Figure 6. Convergence history of GMRES algorithms for system 1 on the missile example. examples in monostatic RCS computation. The convergence curves of the GMRES, SSOR preconditioned GMRES (SSOR-GMRES), innerouter algorithms are included for comparison. For the box-plate examples in bistatic RCS computation with single righthand side, the convergence histories are given in Figure 8. In these computations, the MLFMM was employed for accelerating the computation of the matrix/vector products. The value of relaxation parameter in SSOR-GMRES and SSOR- algorithms is.6,
Progress In Electromagnetics Research, PIER 89, 29 349 Residual Norm 1.E+ 1.E-1 1.E-2 GMRES SSOR-GMRES SSOR- 1.E-3 1 2 3 4 5 6 7 8 9 Number of Matrix-vector Multiplications Figure 7. Convergence history of GMRES algorithms for system 1 on the open-cavity example. Residual Norm 1.E+ 1.E-1 1.E-2 GMRES SSOR-GMRES SSOR- 1.E-3 2 4 6 8 1 Number of Matrix-vector Multiplications Figure 8. Convergence history of GMRES algorithms on the boxplate example. inner iteration number in and SSOR- algorithms is taken to be 1. It can be found that the proposed SSOR- algorithm converges fastest than the other algorithms for all the four examples. As shown in Table 1, the number of matrix-vector multiplication and solution time with the above preconditioned algorithms for are given for system 1 in monostatic RCS computation on open-cavity, missile and open-cone examples, where * refers to no convergence
35 Ding, Chen, and Fan Table 1. Number of matrix-vector multiplications and solution time (in seconds) for open structures. Example Opencone Number of matrix-vector multiplications GMRES 215 SSOR- GMRES 4 SSOR- GMRES SSOR- GMRES Solution time SSOR- 55 187 14s * 3s 1s Missile 3738 >6 1268 378 1358s * 484s 128s Opencavity Boxplate >1 55 63 869 451 41s 617s 342s 224s >1 4 >1 4 7678 2298 * * 183781s 46392s after maximum iterations. The number of matrix-vector multiplication and solution time for the box-plate in bistatic RCS computation are also given in Table 1. It is noted that the number of matrix-vector multiplications (MVM) includes both the number of MVM in the inner iteration and the one in the outer iteration in the and SSOR- algorithms. It can be found that the proposed SSOR- algorithm can save much time than other algorithms. For both open-cavity example and open-cone one, the solution time of the FMGRES algorithm is less than the GMRES. However, the number of matrix-vector multiplications for is more than that of GMRES. This is mainly due to the use of the near-part impedance matrix on the inner iteration of the algorithm. Furthermore, small size of the near-part impedance matrix A N results in the low cost for implementation of matrix-vector multiplications in the inner iteration. When compared with, the improved SSOR- save the solution time by a factor of 3. for the open cone example, 3.8 for the missile, 1.5 for the open cavity and 4. for the box-plate. This demonstrates the efficiency of the newly proposed SSOR preconditioned algorithm for the open structures. A critical question in the use of the SSOR preconditioned innerouter algorithm is to what precision the preconditioner should be solved, i.e., what stopping threshold should be used in the inner iteration. In other words, we wish to balance the inner and outer iterations so that the total number of operations is minimized. In order to answer this question, we vary the inner iteration number from 5 to 75 and the total number of matrix-vector multiplications and CPU time of SSOR- are given in Figure 9for the open cone example. In these computations, the value of relaxation parameter in the SSOR- algorithms is taken to be.6. It can be observed
Progress In Electromagnetics Research, PIER 89, 29 351 Number of Matrix-vector Multiplication 14 12 1 8 6 4 2 Number of MVM Solution Time 3 25 2 15 1 5 CPU Time(s) 5 1 15 2 25 3 35 4 45 5 55 6 65 7 75 Inner Iteration Number Figure 9. Number of matrix-vector multiplications and solution time with SSOR- algorithm for the different inner iteration numbers on the open-cone example. 5 5 Number of Matrix-vector Multiplication 45 4 35 3 25 2 15 1 5 Number of MVM Solution time 45 4 35 3 25 2 15 1 5 CPU Time(s).1.2.3.4.5.6.7.8.9 1 1.1 1.2 Relaxation Parameter Figure 1. Number of matrix-vector multiplications and solution time with SSOR- algorithm for the different relaxation parameters on the open cone example.
352 Ding, Chen, and Fan that the number of matrix-vector multiplications and solution time is almost increased with the increasing of the inner iteration. When the inner iteration number is taken to be 1, the minimal solution time is 1.1 seconds and the number of matrix-vector multiplications is 187. Another important factor of the convergence rate for the proposed SSOR- algorithm is the relaxation parameter. We vary the relaxation parameter from.1 to 1.2 and the total number of matrixvector multiplications (MVM) and solution time of SSOR- are given in Figure 1. In these computations, the inner iteration number in the SSOR- algorithms is taken to be 1. It can be observed that the number of MVM and solution time of SSOR- decrease rapidly when the value of the relaxation parameter is increased from.1 to.3. The total MVM number and solution time is nearly invariant with the value of the relaxation parameter varying from.4 1.. When the value of the relaxation parameter is increased from 1.1 to 1.2, the total MVM number and solution time increases rapidly. When the relaxation parameter is taken to be.8, the minimal number of MVM and the minimal solution time is 176 and 9.9 seconds, respectively. In order to further investigate the performance of the proposed SSOR- algorithm, the number of matrix-vector multiplications (MVM) with and SSOR- algorithms is given for the different incident angles in monostatic RCS computation for 13 Number of matrix-vector multiplication 12 11 1 9 8 7 6 5 4 SSOR- 3 2 4 6 8 1 12 14 16 18 Incident angles Figure 11. Number of matrix-vector multiplications with and SSOR- algorithms for the different incident angles (or systems) on the open cavity.
Progress In Electromagnetics Research, PIER 89, 29 353 Number of matrix-vector multiplication 7 6 5 4 3 2 1 SSOR- 2 4 6 8 1 12 14 16 18 Incident angles Figure 12. Number of matrix-vector multiplications with and SSOR- algorithms for the different incident angles (or systems) on the open cone. Number of matrix-vector multiplicati 14 12 1 8 6 4 2 SSOR- 2 4 6 8 1 12 14 16 18 Incident angles Figure 13. Number of matrix-vector multiplications with and SSOR- algorithms for the different incident angles (or systems) on the missile. open-cone, open-cavity and missile examples in Figures 11 13. The sets of angles of interest for the monstatic RCS of the missile example vary from to 18 degree in ϕ direction when θ is fixed at 9 degree. For both open cavity and open cone examples, the sets of angles of interest for the monstatic RCS vary from to 18 degree in θ direction
354 Ding, Chen, and Fan when ϕ is fixed at degree. The increasing step is one degree and the total number of right-hand sides to be solved for the complete monostatic RCS calculation is 181 for each example. In these computations, the inner iteration number in both and SSOR- algorithms is taken to be 1. The value of relaxation parameter in SSOR- is.6. It can be observed that the SSOR- algorithm can greatly improve the convergence. It can also be found that the number of MVM for varies largely from one system to another for each example. The number of MVM for SSOR- varies slowly on open-cavity example and is almost constant on open-cone example. As shown in Table 2, the cumulated number of matrix-vector multiplications and the total elapsed solution time for a complete monostatic RCS calculation are summarized for each example using and SSOR- algorithms. Depending on the geometry, the overall gain ranges from a factor of 1.6 to 2.7 for both computational time and total number of matrix-vector multiplications. Table 2. Total number of matrix-vector multiplications and solution time (in seconds) for a complete monostatic RCS calculation. Geometry Number of matrix-vector multiplications SSOR- Solution time SSOR- Open cavity 186626 122958 84785 s 5486 s Open -cone 73788 3322 8968 s 344 s Missile 7941 195539 72813 s 398 s 5. CONCLUSIONS AND COMMENTS In this paper, the SSOR-preconditioned inner-outer algorithm is presented for solving EFIE with single and multiple right-hand sides in both bistatic and monostatic RCS calculation for open structures. The symmetric successive over-relaxation (SSOR) preconditioner is constructed based on the near-part matrix of the EFIE in the inner iteration of. The combined effect of the prescribed SSOR preconditioner coupled with inner-outer algorithm is very beneficial for the convergence of the iterative methods. It can be found that the proposed method can reduce both the number of matrix-vector multiplications and the computational
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